Wednesday, September 11, 2013

Hoppe on Euclidean Geometry

Here is a poor reasoning at work:

“Further, the old rationalist claims that Euclidean geometry is a priori yet incorporates empirical knowledge about space becomes supported, too, in view of our insight into the praxeological constraints on knowledge. Since the discovery of non-Euclidean geometries and in particular since Einstein’s relativistic theory of gravitation, the prevailing position regarding geometry is once again empiricist and formalist. It conceives of geometry as either being part of empirical, a posteriori physics, or as being empirically meaningless formalisms. That geometry is either mere play or forever subject to empirical testing seems to be irreconcilable with the fact that Euclidean geometry is the foundation of engineering and construction, and that nobody in those fields ever thinks of such propositions as only hypothetically true.” (Hoppe 2006: 287–288).

On the contrary, that Euclidean geometry is highly useful in certain areas does not refute the epistemological status of applied geometry as synthetic a posteriori. And pure geometry remains analytic a priori and necessarily true and known a priorionly when it is understood as a pure, non-empirical theory.

For how do human beings know that non-Euclidean, curved Riemannian geometry is a better theory of space-time, and that Euclidean geometry is actually only a useful approximation applicable to a certain domain? The answer is: empirically, not a priori.

And the fact that people in “engineering and construction” might never think of Euclidean geometry as “only hypothetically true” or a mere approximation is irrelevant: it commits an appeal to invalid authority.

6 comments:

It is a good thing that a sound philosopher like you takes a critical look at Hoppe, who unfortunately is rather influential in libertarian circles. Thanks for the effort, LK, and I am looking forward to more contributions.

Personally, I am just too impatient to systematically critique Hoppe; I quickly get to the point where I feel offended by his irresponsible use of words. After all, from someone who purports to communicate with me, I expect a demonstrable effort at intelligibility.

Am I wrong? The first sentence that you quote does not even seem to be a proper English sentence.

Haha, so Hoppe's idea is that because experts who make regular use of Euclidean geometry assume it to be true in their figures, they must be unable to distinguish between truth as necessary vs. contingent? That's amazing. It's like the negation of an appeal to anonymous authority — an appeal to personal incredulity inspired by the assumed ineptitude of an unnamed, homogeneous mass of experts. I've never seen its like before.

"For how do human beings know that non-Euclidean, curved Riemannian geometry is a better theory of space-time, and that Euclidean geometry is actually only a useful approximation applicable to a certain domain? The answer is: empirically, not a priori."

Empirical experiments utilize equipment and tools that are produced on the basis of Euclidean geometry. The dimensions are Euclidean. The equipment and its relation to the researcher is assumed as Euclidean. The relationship of the equipment to the researcher to the data collection is Euclidean.

One can't claim that using Euclidean geometry falsifies Euclidean geometry.

The discovery of non-Euclidean geometry in cosmology does not in any way falsify the synthetic a priori truth of Euclidean geometry. Non-Euclidean geometry is BUILT on Euclidean geometry. You can't have one without the other. Both are necessary.

What Kant proved is that our minds are a priori Euclidean AND non-Euclidean, although he did not make the non-Euclidean aspect explicit.

"Empirical experiments utilize equipment and tools that are produced on the basis of Euclidean geometry. "

Irrelevant to the question of whether pure Euclidean geometry provides a necessary universally true theory of space known a priori.

Your absurd comment is like saying that, since many instruments are built with Newtonian mechanics, Newtonian mechanics must be a universally true theory of the universe known a priori. Well, Newtonian mechanics isn't: it is true of a limited domain, just as Euclidean geometry, And we known this empirically.

"What Kant proved is that our minds are a priori Euclidean AND non-Euclidean,"

Mike, this is simply wrong. You are being misled by history. You can indeed construct nonE examples in E space, and that is normally how people first meet nonE geometry. But the situation is entirely symmetric. You could start with some nonE geometry.Geometry in mathematics is defined formally in terms of a "space" and transformations of it. No particular form has special status.Google Klein program geometry for more.